Why are the two tangents drawn from an external point to a circle equal in length?

Let P be the external point and let the two tangents touch the circle at A and B, with O the centre. Join OA, OB and OP. Since a tangent is perpendicular to the radius at the point of contact, angles OAP and OBP are both 90°. Now compare triangles OAP and OBP: OA = OB (both are radii), OP is common to both, and both have a right angle. By the RHS congruence rule (right angle, hypotenuse OP, side radius), the two triangles are congruent. Therefore the corresponding sides PA and PB are equal. So the two tangent segments from an external point are always equal in length. This is why tangent lengths in a quadrilateral circumscribing a circle satisfy neat equal-pair relationships.